﻿#region " Disclaimer"
//************************************************************************************
// BigInteger Class Version 1.03
//
// Copyright (c) 2002 Chew Keong TAN
// All rights reserved.
//
// Permission is hereby granted, free of charge, to any person obtaining a
// copy of this software and associated documentation files (the
// "Software"), to deal in the Software without restriction, including
// without limitation the rights to use, copy, modify, merge, publish,
// distribute, and/or sell copies of the Software, and to permit persons
// to whom the Software is furnished to do so, provided that the above
// copyright notice(s) and this permission notice appear in all copies of
// the Software and that both the above copyright notice(s) and this
// permission notice appear in supporting documentation.
//
// THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
// OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
// MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT
// OF THIRD PARTY RIGHTS. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR
// HOLDERS INCLUDED IN THIS NOTICE BE LIABLE FOR ANY CLAIM, OR ANY SPECIAL
// INDIRECT OR CONSEQUENTIAL DAMAGES, OR ANY DAMAGES WHATSOEVER RESULTING
// FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT,
// NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION
// WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
//************************************************************************************
#endregion

using System;

namespace Utils
{
	public class BigInteger
	{

		// maximum length of the BigInteger in uint (4 bytes)
		// change this to suit the required level of precision.
		private const int maxLength = 300;

		// primes smaller than 2000 to test the generated prime number
		internal static readonly int[] primesBelow2000 = {
					2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
					101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199,
					211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293,
					307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397,
					401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499,
					503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599,
					601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691,
					701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797,
					809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887,
					907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997,
					1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097,
					1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193,
					1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297,
					1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399,
					1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499,
					1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597,
					1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699,
					1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789,
					1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889,
					1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999 };


		private uint[] data = null;             // stores bytes from the Big Integer
		internal int dataLength;                 // number of actual chars used

		#region " CONSTRUCTORS "
		//***********************************************************************
		// Constructor (Default value for BigInteger is 0
		//***********************************************************************

		public BigInteger()
		{
			data = new uint[maxLength];
			dataLength = 1;
		}


		//***********************************************************************
		// Constructor (Default value provided by long)
		//***********************************************************************

		public BigInteger(long value)
		{
			data = new uint[maxLength];
			long tempVal = value;

			// copy bytes from long to BigInteger without any assumption of
			// the length of the long datatype

			dataLength = 0;
			while (value != 0 && dataLength < maxLength)
			{
				data[dataLength] = (uint)(value & 0xFFFFFFFF);
				value >>= 32;
				dataLength++;
			}

			if (tempVal > 0)         // overflow check for +ve value
			{
				if (value != 0 || (data[maxLength - 1] & 0x80000000) != 0)
					throw (new ArithmeticException("Positive overflow in constructor."));
			}
			else if (tempVal < 0)    // underflow check for -ve value
			{
				if (value != -1 || (data[dataLength - 1] & 0x80000000) == 0)
					throw (new ArithmeticException("Negative underflow in constructor."));
			}

			if (dataLength == 0)
				dataLength = 1;
		}


		//***********************************************************************
		// Constructor (Default value provided by ulong)
		//***********************************************************************

		public BigInteger(ulong value)
		{
			data = new uint[maxLength];

			// copy bytes from ulong to BigInteger without any assumption of
			// the length of the ulong datatype

			dataLength = 0;
			while (value != 0 && dataLength < maxLength)
			{
				data[dataLength] = (uint)(value & 0xFFFFFFFF);
				value >>= 32;
				dataLength++;
			}

			if (value != 0 || (data[maxLength - 1] & 0x80000000) != 0)
				throw (new ArithmeticException("Positive overflow in constructor."));

			if (dataLength == 0)
				dataLength = 1;
		}



		//***********************************************************************
		// Constructor (Default value provided by BigInteger)
		//***********************************************************************

		public BigInteger(BigInteger bi)
		{
			data = new uint[maxLength];

			dataLength = bi.dataLength;

			for (int i = 0; i < dataLength; i++)
				data[i] = bi.data[i];
		}


		//***********************************************************************
		// Constructor (Default value provided by a string of digits of the
		//              specified base)
		//
		// Example (base 10)
		// -----------------
		// To initialize "a" with the default value of 1234 in base 10
		//      BigInteger a = new BigInteger("1234", 10)
		//
		// To initialize "a" with the default value of -1234
		//      BigInteger a = new BigInteger("-1234", 10)
		//
		// Example (base 16)
		// -----------------
		// To initialize "a" with the default value of 0x1D4F in base 16
		//      BigInteger a = new BigInteger("1D4F", 16)
		//
		// To initialize "a" with the default value of -0x1D4F
		//      BigInteger a = new BigInteger("-1D4F", 16)
		//
		// Note that string values are specified in the <sign><magnitude>
		// format.
		//
		//***********************************************************************

		public BigInteger(string value, int radix)
		{
			BigInteger multiplier = new BigInteger(1);
			BigInteger result = new BigInteger();
			value = (value.ToUpper()).Trim();
			int limit = 0;

			if (value[0] == '-')
				limit = 1;

			for (int i = value.Length - 1; i >= limit; i--)
			{
				int posVal = (int)value[i];

				if (posVal >= '0' && posVal <= '9')
					posVal -= '0';
				else if (posVal >= 'A' && posVal <= 'Z')
					posVal = (posVal - 'A') + 10;
				else
					posVal = 9999999;       // arbitrary large


				if (posVal >= radix)
					throw (new ArithmeticException("Invalid string in constructor."));
				else
				{
					if (value[0] == '-')
						posVal = -posVal;

					result = result + (multiplier * posVal);

					if ((i - 1) >= limit)
						multiplier = multiplier * radix;
				}
			}

			if (value[0] == '-')     // negative values
			{
				if ((result.data[maxLength - 1] & 0x80000000) == 0)
					throw (new ArithmeticException("Negative underflow in constructor."));
			}
			else    // positive values
			{
				if ((result.data[maxLength - 1] & 0x80000000) != 0)
					throw (new ArithmeticException("Positive overflow in constructor."));
			}

			data = new uint[maxLength];
			for (int i = 0; i < result.dataLength; i++)
				data[i] = result.data[i];

			dataLength = result.dataLength;
		}


		//***********************************************************************
		// Constructor (Default value provided by an array of bytes)
		//
		// The lowest index of the input byte array (i.e [0]) should contain the
		// most significant byte of the number, and the highest index should
		// contain the least significant byte.
		//
		// E.g.
		// To initialize "a" with the default value of 0x1D4F in base 16
		//      byte[] temp = { 0x1D, 0x4F };
		//      BigInteger a = new BigInteger(temp)
		//
		// Note that this method of initialization does not allow the
		// sign to be specified.
		//
		//***********************************************************************

		public BigInteger(byte[] inData)
		{
			dataLength = inData.Length >> 2;

			int leftOver = inData.Length & 0x3;
			if (leftOver != 0)         // length not multiples of 4
				dataLength++;


			if (dataLength > maxLength)
				throw (new ArithmeticException("Byte overflow in constructor."));

			data = new uint[maxLength];

			for (int i = inData.Length - 1, j = 0; i >= 3; i -= 4, j++)
			{
				//data[j] = (uint)((inData[i - 3] << 24) + (inData[i - 2] << 16) +
				//				 (inData[i - 1] << 8) + inData[i]);
				data[j] = ((uint)(inData[i - 3]) << 24) + ((uint)(inData[i - 2]) << 16) + ((uint)(inData[i - 1] << 8)) + ((uint)(inData[i]));
			}

			if (leftOver == 1)
				data[dataLength - 1] = (uint)inData[0];
			else if (leftOver == 2)
				data[dataLength - 1] = (uint)((inData[0] << 8) + inData[1]);
			else if (leftOver == 3)
				data[dataLength - 1] = (uint)((inData[0] << 16) + (inData[1] << 8) + inData[2]);


			while (dataLength > 1 && data[dataLength - 1] == 0)
				dataLength--;

			//Console.WriteLine("Len = " + dataLength);
		}


		//***********************************************************************
		// Constructor (Default value provided by an array of bytes of the
		// specified length.)
		//***********************************************************************

		public BigInteger(byte[] inData, int inLen)
		{
			dataLength = inLen >> 2;

			int leftOver = inLen & 0x3;
			if (leftOver != 0)         // length not multiples of 4
				dataLength++;

			if (dataLength > maxLength || inLen > inData.Length)
				throw (new ArithmeticException("Byte overflow in constructor."));


			data = new uint[maxLength];

			for (int i = inLen - 1, j = 0; i >= 3; i -= 4, j++)
			{
				data[j] = (uint)((inData[i - 3] << 24) + (inData[i - 2] << 16) +
								 (inData[i - 1] << 8) + inData[i]);
			}

			if (leftOver == 1)
				data[dataLength - 1] = (uint)inData[0];
			else if (leftOver == 2)
				data[dataLength - 1] = (uint)((inData[0] << 8) + inData[1]);
			else if (leftOver == 3)
				data[dataLength - 1] = (uint)((inData[0] << 16) + (inData[1] << 8) + inData[2]);


			if (dataLength == 0)
				dataLength = 1;

			while (dataLength > 1 && data[dataLength - 1] == 0)
				dataLength--;

			//Console.WriteLine("Len = " + dataLength);
		}


		//***********************************************************************
		// Constructor (Default value provided by an array of unsigned integers)
		//*********************************************************************
		public BigInteger(uint[] inData)
		{
			dataLength = inData.Length;

			if (dataLength > maxLength)
				throw (new ArithmeticException("Byte overflow in constructor."));

			data = new uint[maxLength];

			for (int i = dataLength - 1, j = 0; i >= 0; i--, j++)
				data[j] = inData[i];

			while (dataLength > 1 && data[dataLength - 1] == 0)
				dataLength--;

			//Console.WriteLine("Len = " + dataLength);
		}
		#endregion

		#region " OPERATOR OVERLOADS "

		//***********************************************************************
		// Overloading of the typecast operator.
		// For BigInteger bi = 10;
		//***********************************************************************
		public static implicit operator BigInteger(long value)
		{
			return (new BigInteger(value));
		}

		public static implicit operator BigInteger(ulong value)
		{
			return (new BigInteger(value));
		}

		public static implicit operator BigInteger(int value)
		{
			return (new BigInteger((long)value));
		}

		public static implicit operator BigInteger(uint value)
		{
			return (new BigInteger((ulong)value));
		}


		//***********************************************************************
		// Overloading of addition operator
		//***********************************************************************
		public static BigInteger operator +(BigInteger bi1, BigInteger bi2)
		{
			BigInteger result = new BigInteger();

			result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

			long carry = 0;
			for (int i = 0; i < result.dataLength; i++)
			{
				long sum = (long)bi1.data[i] + (long)bi2.data[i] + carry;
				carry = sum >> 32;
				result.data[i] = (uint)(sum & 0xFFFFFFFF);
			}

			if (carry != 0 && result.dataLength < maxLength)
			{
				result.data[result.dataLength] = (uint)(carry);
				result.dataLength++;
			}

			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;


			// overflow check
			int lastPos = maxLength - 1;
			if ((bi1.data[lastPos] & 0x80000000) == (bi2.data[lastPos] & 0x80000000) &&
			   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
			{
				throw (new ArithmeticException());
			}

			return result;
		}


		//***********************************************************************
		// Overloading of the unary ++ operator
		//***********************************************************************
		public static BigInteger operator ++(BigInteger bi1)
		{
			BigInteger result = new BigInteger(bi1);

			long val, carry = 1;
			int index = 0;

			while (carry != 0 && index < maxLength)
			{
				val = (long)(result.data[index]);
				val++;

				result.data[index] = (uint)(val & 0xFFFFFFFF);
				carry = val >> 32;

				index++;
			}

			if (index > result.dataLength)
				result.dataLength = index;
			else
			{
				while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
					result.dataLength--;
			}

			// overflow check
			int lastPos = maxLength - 1;

			// overflow if initial value was +ve but ++ caused a sign
			// change to negative.

			if ((bi1.data[lastPos] & 0x80000000) == 0 &&
			   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
			{
				throw (new ArithmeticException("Overflow in ++."));
			}
			return result;
		}


		//***********************************************************************
		// Overloading of subtraction operator
		//***********************************************************************
		public static BigInteger operator -(BigInteger bi1, BigInteger bi2)
		{
			BigInteger result = new BigInteger();

			result.dataLength = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

			long carryIn = 0;
			for (int i = 0; i < result.dataLength; i++)
			{
				long diff;

				diff = (long)bi1.data[i] - (long)bi2.data[i] - carryIn;
				result.data[i] = (uint)(diff & 0xFFFFFFFF);

				if (diff < 0)
					carryIn = 1;
				else
					carryIn = 0;
			}

			// roll over to negative
			if (carryIn != 0)
			{
				for (int i = result.dataLength; i < maxLength; i++)
					result.data[i] = 0xFFFFFFFF;
				result.dataLength = maxLength;
			}

			// fixed in v1.03 to give correct datalength for a - (-b)
			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;

			// overflow check

			int lastPos = maxLength - 1;
			if ((bi1.data[lastPos] & 0x80000000) != (bi2.data[lastPos] & 0x80000000) &&
			   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
			{
				throw (new ArithmeticException());
			}

			return result;
		}


		//***********************************************************************
		// Overloading of the unary -- operator
		//***********************************************************************
		public static BigInteger operator --(BigInteger bi1)
		{
			BigInteger result = new BigInteger(bi1);

			long val;
			bool carryIn = true;
			int index = 0;

			while (carryIn && index < maxLength)
			{
				val = (long)(result.data[index]);
				val--;

				result.data[index] = (uint)(val & 0xFFFFFFFF);

				if (val >= 0)
					carryIn = false;

				index++;
			}

			if (index > result.dataLength)
				result.dataLength = index;

			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;

			// overflow check
			int lastPos = maxLength - 1;

			// overflow if initial value was -ve but -- caused a sign
			// change to positive.

			if ((bi1.data[lastPos] & 0x80000000) != 0 &&
			   (result.data[lastPos] & 0x80000000) != (bi1.data[lastPos] & 0x80000000))
			{
				throw (new ArithmeticException("Underflow in --."));
			}

			return result;
		}


		//***********************************************************************
		// Overloading of multiplication operator
		//***********************************************************************

		public static BigInteger operator *(BigInteger bi1, BigInteger bi2)
		{
			int lastPos = maxLength - 1;
			bool bi1Neg = false, bi2Neg = false;

			// take the absolute value of the inputs
			try
			{
				if ((bi1.data[lastPos] & 0x80000000) != 0)     // bi1 negative
				{
					bi1Neg = true; bi1 = -bi1;
				}
				if ((bi2.data[lastPos] & 0x80000000) != 0)     // bi2 negative
				{
					bi2Neg = true; bi2 = -bi2;
				}
			}
			catch (Exception) { }

			BigInteger result = new BigInteger();

			// multiply the absolute values
			try
			{
				for (int i = 0; i < bi1.dataLength; i++)
				{
					if (bi1.data[i] == 0) continue;

					ulong mcarry = 0;
					for (int j = 0, k = i; j < bi2.dataLength; j++, k++)
					{
						// k = i + j
						ulong val = ((ulong)bi1.data[i] * (ulong)bi2.data[j]) +
									 (ulong)result.data[k] + mcarry;

						result.data[k] = (uint)(val & 0xFFFFFFFF);
						mcarry = (val >> 32);
					}

					if (mcarry != 0)
						result.data[i + bi2.dataLength] = (uint)mcarry;
				}
			}
			catch (Exception)
			{
				throw (new ArithmeticException("Multiplication overflow."));
			}


			result.dataLength = bi1.dataLength + bi2.dataLength;
			if (result.dataLength > maxLength)
				result.dataLength = maxLength;

			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;

			// overflow check (result is -ve)
			if ((result.data[lastPos] & 0x80000000) != 0)
			{
				if (bi1Neg != bi2Neg && result.data[lastPos] == 0x80000000)    // different sign
				{
					// handle the special case where multiplication produces
					// a max negative number in 2's complement.

					if (result.dataLength == 1)
						return result;
					else
					{
						bool isMaxNeg = true;
						for (int i = 0; i < result.dataLength - 1 && isMaxNeg; i++)
						{
							if (result.data[i] != 0)
								isMaxNeg = false;
						}

						if (isMaxNeg)
							return result;
					}
				}

				throw (new ArithmeticException("Multiplication overflow."));
			}

			// if input has different signs, then result is -ve
			if (bi1Neg != bi2Neg)
				return -result;

			return result;
		}

		//***********************************************************************
		// Overloading of unary << operators
		//***********************************************************************

		public static BigInteger operator <<(BigInteger bi1, int shiftVal)
		{
			BigInteger result = new BigInteger(bi1);
			result.dataLength = shiftLeft(result.data, shiftVal);

			return result;
		}


		// least significant bits at lower part of buffer

		private static int shiftLeft(uint[] buffer, int shiftVal)
		{
			int shiftAmount = 32;
			int bufLen = buffer.Length;

			while (bufLen > 1 && buffer[bufLen - 1] == 0)
				bufLen--;

			for (int count = shiftVal; count > 0; )
			{
				if (count < shiftAmount)
					shiftAmount = count;

				//Console.WriteLine("shiftAmount = {0}", shiftAmount);

				ulong carry = 0;
				for (int i = 0; i < bufLen; i++)
				{
					ulong val = ((ulong)buffer[i]) << shiftAmount;
					val |= carry;

					buffer[i] = (uint)(val & 0xFFFFFFFF);
					carry = val >> 32;
				}

				if (carry != 0)
				{
					if (bufLen + 1 <= buffer.Length)
					{
						buffer[bufLen] = (uint)carry;
						bufLen++;
					}
				}
				count -= shiftAmount;
			}
			return bufLen;
		}


		//***********************************************************************
		// Overloading of unary >> operators
		//***********************************************************************

		public static BigInteger operator >>(BigInteger bi1, int shiftVal)
		{
			BigInteger result = new BigInteger(bi1);
			result.dataLength = shiftRight(result.data, shiftVal);


			if ((bi1.data[maxLength - 1] & 0x80000000) != 0) // negative
			{
				for (int i = maxLength - 1; i >= result.dataLength; i--)
					result.data[i] = 0xFFFFFFFF;

				uint mask = 0x80000000;
				for (int i = 0; i < 32; i++)
				{
					if ((result.data[result.dataLength - 1] & mask) != 0)
						break;

					result.data[result.dataLength - 1] |= mask;
					mask >>= 1;
				}
				result.dataLength = maxLength;
			}

			return result;
		}


		private static int shiftRight(uint[] buffer, int shiftVal)
		{
			int shiftAmount = 32;
			int invShift = 0;
			int bufLen = buffer.Length;

			while (bufLen > 1 && buffer[bufLen - 1] == 0)
				bufLen--;

			//Console.WriteLine("bufLen = " + bufLen + " buffer.Length = " + buffer.Length);

			for (int count = shiftVal; count > 0; )
			{
				if (count < shiftAmount)
				{
					shiftAmount = count;
					invShift = 32 - shiftAmount;
				}

				//Console.WriteLine("shiftAmount = {0}", shiftAmount);

				ulong carry = 0;
				for (int i = bufLen - 1; i >= 0; i--)
				{
					ulong val = ((ulong)buffer[i]) >> shiftAmount;
					val |= carry;

					//carry = ((ulong)buffer[i]) << invShift;
					carry = (((ulong)buffer[i]) << invShift) & 0xFFFFFFFF;
					buffer[i] = (uint)(val);
				}

				count -= shiftAmount;
			}

			while (bufLen > 1 && buffer[bufLen - 1] == 0)
				bufLen--;

			return bufLen;
		}


		//***********************************************************************
		// Overloading of the NOT operator (1's complement)
		//***********************************************************************

		public static BigInteger operator ~(BigInteger bi1)
		{
			BigInteger result = new BigInteger(bi1);

			for (int i = 0; i < maxLength; i++)
				result.data[i] = (uint)(~(bi1.data[i]));

			result.dataLength = maxLength;

			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;

			return result;
		}

		//***********************************************************************
		// Overloading of the NEGATE operator (2's complement)
		//***********************************************************************

		public static BigInteger operator -(BigInteger bi1)
		{
			// handle neg of zero separately since it'll cause an overflow
			// if we proceed.

			if (bi1.dataLength == 1 && bi1.data[0] == 0)
				return (new BigInteger());

			BigInteger result = new BigInteger(bi1);

			// 1's complement
			for (int i = 0; i < maxLength; i++)
				result.data[i] = (uint)(~(bi1.data[i]));

			// add one to result of 1's complement
			long val, carry = 1;
			int index = 0;

			while (carry != 0 && index < maxLength)
			{
				val = (long)(result.data[index]);
				val++;

				result.data[index] = (uint)(val & 0xFFFFFFFF);
				carry = val >> 32;

				index++;
			}

			if ((bi1.data[maxLength - 1] & 0x80000000) == (result.data[maxLength - 1] & 0x80000000))
				throw (new ArithmeticException("Overflow in negation.\n"));

			result.dataLength = maxLength;

			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;
			return result;
		}


		//***********************************************************************
		// Overloading of equality operator
		//***********************************************************************

		public static bool operator ==(BigInteger bi1, BigInteger bi2)
		{
			return bi1.Equals(bi2);
		}

		public static bool operator !=(BigInteger bi1, BigInteger bi2)
		{
			return !(bi1.Equals(bi2));
		}

		public override bool Equals(object o)
		{
			if (o == null)
				return false;

			BigInteger bi = (BigInteger)o;

			if (this.dataLength != bi.dataLength)
				return false;

			for (int i = 0; i < this.dataLength; i++)
			{
				if (this.data[i] != bi.data[i])
					return false;
			}
			return true;
		}

		public override int GetHashCode()
		{
			return this.ToString().GetHashCode();
		}


		//***********************************************************************
		// Overloading of inequality operator
		//***********************************************************************

		public static bool operator >(BigInteger bi1, BigInteger bi2)
		{
			int pos = maxLength - 1;

			// bi1 is negative, bi2 is positive
			if ((bi1.data[pos] & 0x80000000) != 0 && (bi2.data[pos] & 0x80000000) == 0)
				return false;

				// bi1 is positive, bi2 is negative
			else if ((bi1.data[pos] & 0x80000000) == 0 && (bi2.data[pos] & 0x80000000) != 0)
				return true;

			// same sign
			int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
			for (pos = len - 1; pos >= 0 && bi1.data[pos] == bi2.data[pos]; pos--) ;

			if (pos >= 0)
			{
				if (bi1.data[pos] > bi2.data[pos])
					return true;
				return false;
			}
			return false;
		}

		public static bool operator <(BigInteger bi1, BigInteger bi2)
		{
			int pos = maxLength - 1;

			// bi1 is negative, bi2 is positive
			if ((bi1.data[pos] & 0x80000000) != 0 && (bi2.data[pos] & 0x80000000) == 0)
				return true;

				// bi1 is positive, bi2 is negative
			else if ((bi1.data[pos] & 0x80000000) == 0 && (bi2.data[pos] & 0x80000000) != 0)
				return false;

			// same sign
			int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;
			for (pos = len - 1; pos >= 0 && bi1.data[pos] == bi2.data[pos]; pos--) ;

			if (pos >= 0)
			{
				if (bi1.data[pos] < bi2.data[pos])
					return true;
				return false;
			}
			return false;
		}

		public static bool operator >=(BigInteger bi1, BigInteger bi2)
		{
			return (bi1 == bi2 || bi1 > bi2);
		}

		public static bool operator <=(BigInteger bi1, BigInteger bi2)
		{
			return (bi1 == bi2 || bi1 < bi2);
		}

		#endregion

		//***********************************************************************
		// Private function that supports the division of two numbers with
		// a divisor that has more than 1 digit.
		//
		// Algorithm taken from [1]
		//***********************************************************************

		private static void multiByteDivide(BigInteger bi1, BigInteger bi2,
											BigInteger outQuotient, BigInteger outRemainder)
		{
			uint[] result = new uint[maxLength];

			int remainderLen = bi1.dataLength + 1;
			uint[] remainder = new uint[remainderLen];

			uint mask = 0x80000000;
			uint val = bi2.data[bi2.dataLength - 1];
			int shift = 0, resultPos = 0;

			while (mask != 0 && (val & mask) == 0)
			{
				shift++; mask >>= 1;
			}

			for (int i = 0; i < bi1.dataLength; i++)
				remainder[i] = bi1.data[i];
			shiftLeft(remainder, shift);
			bi2 = bi2 << shift;

			int j = remainderLen - bi2.dataLength;
			int pos = remainderLen - 1;

			ulong firstDivisorByte = bi2.data[bi2.dataLength - 1];
			ulong secondDivisorByte = bi2.data[bi2.dataLength - 2];

			int divisorLen = bi2.dataLength + 1;
			uint[] dividendPart = new uint[divisorLen];

			while (j > 0)
			{
				ulong dividend = ((ulong)remainder[pos] << 32) + (ulong)remainder[pos - 1];

				ulong q_hat = dividend / firstDivisorByte;
				ulong r_hat = dividend % firstDivisorByte;

				bool done = false;
				while (!done)
				{
					done = true;

					if (q_hat == 0x100000000 ||
					   (q_hat * secondDivisorByte) > ((r_hat << 32) + remainder[pos - 2]))
					{
						q_hat--;
						r_hat += firstDivisorByte;

						if (r_hat < 0x100000000)
							done = false;
					}
				}

				for (int h = 0; h < divisorLen; h++)
					dividendPart[h] = remainder[pos - h];

				BigInteger kk = new BigInteger(dividendPart);
				BigInteger ss = bi2 * (long)q_hat;

				while (ss > kk)
				{
					q_hat--;
					ss -= bi2;
				}
				BigInteger yy = kk - ss;

				for (int h = 0; h < divisorLen; h++)
					remainder[pos - h] = yy.data[bi2.dataLength - h];

				result[resultPos++] = (uint)q_hat;

				pos--;
				j--;
			}

			outQuotient.dataLength = resultPos;
			int y = 0;
			for (int x = outQuotient.dataLength - 1; x >= 0; x--, y++)
				outQuotient.data[y] = result[x];
			for (; y < maxLength; y++)
				outQuotient.data[y] = 0;

			while (outQuotient.dataLength > 1 && outQuotient.data[outQuotient.dataLength - 1] == 0)
				outQuotient.dataLength--;

			if (outQuotient.dataLength == 0)
				outQuotient.dataLength = 1;

			outRemainder.dataLength = shiftRight(remainder, shift);

			for (y = 0; y < outRemainder.dataLength; y++)
				outRemainder.data[y] = remainder[y];
			for (; y < maxLength; y++)
				outRemainder.data[y] = 0;

		}


		//***********************************************************************
		// Private function that supports the division of two numbers with
		// a divisor that has only 1 digit.
		//***********************************************************************

		private static void singleByteDivide(BigInteger bi1, BigInteger bi2,
											 BigInteger outQuotient, BigInteger outRemainder)
		{
			uint[] result = new uint[maxLength];
			int resultPos = 0;

			// copy dividend to reminder
			for (int i = 0; i < maxLength; i++)
				outRemainder.data[i] = bi1.data[i];
			outRemainder.dataLength = bi1.dataLength;

			while (outRemainder.dataLength > 1 && outRemainder.data[outRemainder.dataLength - 1] == 0)
				outRemainder.dataLength--;

			ulong divisor = (ulong)bi2.data[0];
			int pos = outRemainder.dataLength - 1;
			ulong dividend = (ulong)outRemainder.data[pos];

			if (dividend >= divisor)
			{
				ulong quotient = dividend / divisor;
				result[resultPos++] = (uint)quotient;

				outRemainder.data[pos] = (uint)(dividend % divisor);
			}
			pos--;

			while (pos >= 0)
			{
				dividend = ((ulong)outRemainder.data[pos + 1] << 32) + (ulong)outRemainder.data[pos];
				ulong quotient = dividend / divisor;
				result[resultPos++] = (uint)quotient;

				outRemainder.data[pos + 1] = 0;
				outRemainder.data[pos--] = (uint)(dividend % divisor);
			}

			outQuotient.dataLength = resultPos;
			int j = 0;
			for (int i = outQuotient.dataLength - 1; i >= 0; i--, j++)
				outQuotient.data[j] = result[i];
			for (; j < maxLength; j++)
				outQuotient.data[j] = 0;

			while (outQuotient.dataLength > 1 && outQuotient.data[outQuotient.dataLength - 1] == 0)
				outQuotient.dataLength--;

			if (outQuotient.dataLength == 0)
				outQuotient.dataLength = 1;

			while (outRemainder.dataLength > 1 && outRemainder.data[outRemainder.dataLength - 1] == 0)
				outRemainder.dataLength--;
		}


		//***********************************************************************
		// Overloading of division operator
		//***********************************************************************

		public static BigInteger operator /(BigInteger bi1, BigInteger bi2)
		{
			BigInteger quotient = new BigInteger();
			BigInteger remainder = new BigInteger();

			int lastPos = maxLength - 1;
			bool divisorNeg = false, dividendNeg = false;

			if ((bi1.data[lastPos] & 0x80000000) != 0)     // bi1 negative
			{
				bi1 = -bi1;
				dividendNeg = true;
			}
			if ((bi2.data[lastPos] & 0x80000000) != 0)     // bi2 negative
			{
				bi2 = -bi2;
				divisorNeg = true;
			}

			if (bi1 < bi2)
			{
				return quotient;
			}

			else
			{
				if (bi2.dataLength == 1)
					singleByteDivide(bi1, bi2, quotient, remainder);
				else
					multiByteDivide(bi1, bi2, quotient, remainder);

				if (dividendNeg != divisorNeg)
					return -quotient;

				return quotient;
			}
		}


		//***********************************************************************
		// Overloading of modulus operator
		//***********************************************************************

		public static BigInteger operator %(BigInteger bi1, BigInteger bi2)
		{
			BigInteger quotient = new BigInteger();
			BigInteger remainder = new BigInteger(bi1);

			int lastPos = maxLength - 1;
			bool dividendNeg = false;

			if ((bi1.data[lastPos] & 0x80000000) != 0)     // bi1 negative
			{
				bi1 = -bi1;
				dividendNeg = true;
			}
			if ((bi2.data[lastPos] & 0x80000000) != 0)     // bi2 negative
				bi2 = -bi2;

			if (bi1 < bi2)
			{
				return remainder;
			}

			else
			{
				if (bi2.dataLength == 1)
					singleByteDivide(bi1, bi2, quotient, remainder);
				else
					multiByteDivide(bi1, bi2, quotient, remainder);

				if (dividendNeg)
					return -remainder;

				return remainder;
			}
		}


		//***********************************************************************
		// Overloading of bitwise AND operator
		//***********************************************************************

		public static BigInteger operator &(BigInteger bi1, BigInteger bi2)
		{
			BigInteger result = new BigInteger();

			int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

			for (int i = 0; i < len; i++)
			{
				uint sum = (uint)(bi1.data[i] & bi2.data[i]);
				result.data[i] = sum;
			}

			result.dataLength = maxLength;

			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;

			return result;
		}


		//***********************************************************************
		// Overloading of bitwise OR operator
		//***********************************************************************

		public static BigInteger operator |(BigInteger bi1, BigInteger bi2)
		{
			BigInteger result = new BigInteger();

			int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

			for (int i = 0; i < len; i++)
			{
				uint sum = (uint)(bi1.data[i] | bi2.data[i]);
				result.data[i] = sum;
			}

			result.dataLength = maxLength;

			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;

			return result;
		}


		//***********************************************************************
		// Overloading of bitwise XOR operator
		//***********************************************************************

		public static BigInteger operator ^(BigInteger bi1, BigInteger bi2)
		{
			BigInteger result = new BigInteger();

			int len = (bi1.dataLength > bi2.dataLength) ? bi1.dataLength : bi2.dataLength;

			for (int i = 0; i < len; i++)
			{
				uint sum = (uint)(bi1.data[i] ^ bi2.data[i]);
				result.data[i] = sum;
			}

			result.dataLength = maxLength;

			while (result.dataLength > 1 && result.data[result.dataLength - 1] == 0)
				result.dataLength--;

			return result;
		}


		//***********************************************************************
		// Returns max(this, bi)
		//***********************************************************************

		internal BigInteger max(BigInteger bi)
		{
			if (this > bi)
				return (new BigInteger(this));
			else
				return (new BigInteger(bi));
		}


		//***********************************************************************
		// Returns min(this, bi)
		//***********************************************************************

		internal BigInteger min(BigInteger bi)
		{
			if (this < bi)
				return (new BigInteger(this));
			else
				return (new BigInteger(bi));

		}


		//***********************************************************************
		// Returns the absolute value
		//***********************************************************************

		internal BigInteger abs()
		{
			if ((this.data[maxLength - 1] & 0x80000000) != 0)
				return (-this);
			else
				return (new BigInteger(this));
		}


		//***********************************************************************
		// Returns a string representing the BigInteger in base 10.
		//***********************************************************************

		public override string ToString()
		{
			return ToString(10);
		}


		//***********************************************************************
		// Returns a string representing the BigInteger in sign-and-magnitude
		// format in the specified radix.
		//
		// Example
		// -------
		// If the value of BigInteger is -255 in base 10, then
		// ToString(16) returns "-FF"
		//
		//***********************************************************************

		internal string ToString(int radix)
		{
			if (radix < 2 || radix > 36)
				throw (new ArgumentException("Radix must be >= 2 and <= 36"));

			string charSet = "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
			string result = "";

			BigInteger a = this;

			bool negative = false;
			if ((a.data[maxLength - 1] & 0x80000000) != 0)
			{
				negative = true;
				try
				{
					a = -a;
				}
				catch (Exception) { }
			}

			BigInteger quotient = new BigInteger();
			BigInteger remainder = new BigInteger();
			BigInteger biRadix = new BigInteger(radix);

			if (a.dataLength == 1 && a.data[0] == 0)
				result = "0";
			else
			{
				while (a.dataLength > 1 || (a.dataLength == 1 && a.data[0] != 0))
				{
					singleByteDivide(a, biRadix, quotient, remainder);

					if (remainder.data[0] < 10)
						result = remainder.data[0] + result;
					else
						result = charSet[(int)remainder.data[0] - 10] + result;

					a = quotient;
				}
				if (negative)
					result = "-" + result;
			}

			return result;
		}


		//***********************************************************************
		// Returns a hex string showing the contains of the BigInteger
		//
		// Examples
		// -------
		// 1) If the value of BigInteger is 255 in base 10, then
		//    ToHexString() returns "FF"
		//
		// 2) If the value of BigInteger is -255 in base 10, then
		//    ToHexString() returns ".....FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF01",
		//    which is the 2's complement representation of -255.
		//
		//***********************************************************************

		internal string ToHexString()
		{
			string result = data[dataLength - 1].ToString("X");

			for (int i = dataLength - 2; i >= 0; i--)
			{
				result += data[i].ToString("X8");
			}

			return result;
		}

		//***********************************************************************
		// Modulo Exponentiation
		//***********************************************************************

		internal BigInteger modPow(BigInteger exp, BigInteger n)
		{
			if ((exp.data[maxLength - 1] & 0x80000000) != 0)
				throw (new ArithmeticException("Exponent must be positive."));

			BigInteger resultNum = 1;
			BigInteger tempNum;
			bool thisNegative = false;

			if ((this.data[maxLength - 1] & 0x80000000) != 0)   // negative this
			{
				tempNum = -this % n;
				thisNegative = true;
			}
			else
				tempNum = this % n;  // ensures (tempNum * tempNum) < b^(2k)

			if ((n.data[maxLength - 1] & 0x80000000) != 0)   // negative n
				n = -n;

			// calculate constant = b^(2k) / m
			BigInteger constant = new BigInteger();

			int i = n.dataLength << 1;
			constant.data[i] = 0x00000001;
			constant.dataLength = i + 1;

			constant = constant / n;
			int totalBits = exp.bitCount();
			int count = 0;

			// perform squaring and multiply exponentiation
			for (int pos = 0; pos < exp.dataLength; pos++)
			{
				uint mask = 0x01;

				for (int index = 0; index < 32; index++)
				{
					if ((exp.data[pos] & mask) != 0)
						resultNum = BarrettReduction(resultNum * tempNum, n, constant);

					mask <<= 1;

					tempNum = BarrettReduction(tempNum * tempNum, n, constant);


					if (tempNum.dataLength == 1 && tempNum.data[0] == 1)
					{
						if (thisNegative && (exp.data[0] & 0x1) != 0)    //odd exp
							return -resultNum;
						return resultNum;
					}
					count++;
					if (count == totalBits)
						break;
				}
			}

			if (thisNegative && (exp.data[0] & 0x1) != 0)    //odd exp
				return -resultNum;

			return resultNum;
		}



		//***********************************************************************
		// Fast calculation of modular reduction using Barrett's reduction.
		// Requires x < b^(2k), where b is the base.  In this case, base is
		// 2^32 (uint).
		//
		// Reference [4]
		//***********************************************************************

		private BigInteger BarrettReduction(BigInteger x, BigInteger n, BigInteger constant)
		{
			int k = n.dataLength,
				kPlusOne = k + 1,
				kMinusOne = k - 1;

			BigInteger q1 = new BigInteger();

			// q1 = x / b^(k-1)
			for (int i = kMinusOne, j = 0; i < x.dataLength; i++, j++)
				q1.data[j] = x.data[i];
			q1.dataLength = x.dataLength - kMinusOne;
			if (q1.dataLength <= 0)
				q1.dataLength = 1;


			BigInteger q2 = q1 * constant;
			BigInteger q3 = new BigInteger();

			// q3 = q2 / b^(k+1)
			for (int i = kPlusOne, j = 0; i < q2.dataLength; i++, j++)
				q3.data[j] = q2.data[i];
			q3.dataLength = q2.dataLength - kPlusOne;
			if (q3.dataLength <= 0)
				q3.dataLength = 1;


			// r1 = x mod b^(k+1)
			// i.e. keep the lowest (k+1) words
			BigInteger r1 = new BigInteger();
			int lengthToCopy = (x.dataLength > kPlusOne) ? kPlusOne : x.dataLength;
			for (int i = 0; i < lengthToCopy; i++)
				r1.data[i] = x.data[i];
			r1.dataLength = lengthToCopy;


			// r2 = (q3 * n) mod b^(k+1)
			// partial multiplication of q3 and n

			BigInteger r2 = new BigInteger();
			for (int i = 0; i < q3.dataLength; i++)
			{
				if (q3.data[i] == 0) continue;

				ulong mcarry = 0;
				int t = i;
				for (int j = 0; j < n.dataLength && t < kPlusOne; j++, t++)
				{
					// t = i + j
					ulong val = ((ulong)q3.data[i] * (ulong)n.data[j]) +
								 (ulong)r2.data[t] + mcarry;

					r2.data[t] = (uint)(val & 0xFFFFFFFF);
					mcarry = (val >> 32);
				}

				if (t < kPlusOne)
					r2.data[t] = (uint)mcarry;
			}
			r2.dataLength = kPlusOne;
			while (r2.dataLength > 1 && r2.data[r2.dataLength - 1] == 0)
				r2.dataLength--;

			r1 -= r2;
			if ((r1.data[maxLength - 1] & 0x80000000) != 0)        // negative
			{
				BigInteger val = new BigInteger();
				val.data[kPlusOne] = 0x00000001;
				val.dataLength = kPlusOne + 1;
				r1 += val;
			}

			while (r1 >= n)
				r1 -= n;

			return r1;
		}


		//***********************************************************************
		// Returns gcd(this, bi)
		//***********************************************************************

		internal BigInteger gcd(BigInteger bi)
		{
			BigInteger x;
			BigInteger y;

			if ((data[maxLength - 1] & 0x80000000) != 0)     // negative
				x = -this;
			else
				x = this;

			if ((bi.data[maxLength - 1] & 0x80000000) != 0)     // negative
				y = -bi;
			else
				y = bi;

			BigInteger g = y;

			while (x.dataLength > 1 || (x.dataLength == 1 && x.data[0] != 0))
			{
				g = x;
				x = y % x;
				y = g;
			}

			return g;
		}


		//***********************************************************************
		// Populates "this" with the specified amount of random bits
		//***********************************************************************

		internal void genRandomBits(int bits, Random rand)
		{
			int dwords = bits >> 5;
			int remBits = bits & 0x1F;

			if (remBits != 0)
				dwords++;

			if (dwords > maxLength)
				throw (new ArithmeticException("Number of required bits > maxLength."));

			for (int i = 0; i < dwords; i++)
				data[i] = (uint)(rand.NextDouble() * 0x100000000);

			for (int i = dwords; i < maxLength; i++)
				data[i] = 0;

			if (remBits != 0)
			{
				uint mask = (uint)(0x01 << (remBits - 1));
				data[dwords - 1] |= mask;

				mask = (uint)(0xFFFFFFFF >> (32 - remBits));
				data[dwords - 1] &= mask;
			}
			else
				data[dwords - 1] |= 0x80000000;

			dataLength = dwords;

			if (dataLength == 0)
				dataLength = 1;
		}


		//***********************************************************************
		// Probabilistic prime test based on Fermat's little theorem
		//
		// for any a < p (p does not divide a) if
		//      a^(p-1) mod p != 1 then p is not prime.
		//
		// Otherwise, p is probably prime (pseudoprime to the chosen base).
		//
		// Returns
		// -------
		// True if "this" is a pseudoprime to randomly chosen
		// bases.  The number of chosen bases is given by the "confidence"
		// parameter.
		//
		// False if "this" is definitely NOT prime.
		//
		// Note - this method is fast but fails for Carmichael numbers except
		// when the randomly chosen base is a factor of the number.
		//
		//***********************************************************************

		internal bool FermatLittleTest(int confidence)
		{
			BigInteger thisVal;
			if ((this.data[maxLength - 1] & 0x80000000) != 0)        // negative
				thisVal = -this;
			else
				thisVal = this;

			if (thisVal.dataLength == 1)
			{
				// test small numbers
				if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
					return false;
				else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
					return true;
			}

			if ((thisVal.data[0] & 0x1) == 0)     // even numbers
				return false;

			int bits = thisVal.bitCount();
			BigInteger a = new BigInteger();
			BigInteger p_sub1 = thisVal - (new BigInteger(1));
			Random rand = new Random();

			for (int round = 0; round < confidence; round++)
			{
				bool done = false;

				while (!done)		// generate a < n
				{
					int testBits = 0;

					// make sure "a" has at least 2 bits
					while (testBits < 2)
						testBits = (int)(rand.NextDouble() * bits);

					a.genRandomBits(testBits, rand);

					int byteLen = a.dataLength;

					// make sure "a" is not 0
					if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
						done = true;
				}

				// check whether a factor exists (fix for version 1.03)
				BigInteger gcdTest = a.gcd(thisVal);
				if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
					return false;

				// calculate a^(p-1) mod p
				BigInteger expResult = a.modPow(p_sub1, thisVal);

				int resultLen = expResult.dataLength;

				// is NOT prime is a^(p-1) mod p != 1

				if (resultLen > 1 || (resultLen == 1 && expResult.data[0] != 1))
				{
					return false;
				}
			}

			return true;
		}


		//***********************************************************************
		// Probabilistic prime test based on Rabin-Miller's
		//
		// for any p > 0 with p - 1 = 2^s * t
		//
		// p is probably prime (strong pseudoprime) if for any a < p,
		// 1) a^t mod p = 1 or
		// 2) a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
		//
		// Otherwise, p is composite.
		//
		// Returns
		// -------
		// True if "this" is a strong pseudoprime to randomly chosen
		// bases.  The number of chosen bases is given by the "confidence"
		// parameter.
		//
		// False if "this" is definitely NOT prime.
		//
		//***********************************************************************

		internal bool RabinMillerTest(int confidence)
		{
			BigInteger thisVal;
			if ((this.data[maxLength - 1] & 0x80000000) != 0)        // negative
				thisVal = -this;
			else
				thisVal = this;

			if (thisVal.dataLength == 1)
			{
				// test small numbers
				if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
					return false;
				else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
					return true;
			}

			if ((thisVal.data[0] & 0x1) == 0)     // even numbers
				return false;


			// calculate values of s and t
			BigInteger p_sub1 = thisVal - (new BigInteger(1));
			int s = 0;

			for (int index = 0; index < p_sub1.dataLength; index++)
			{
				uint mask = 0x01;

				for (int i = 0; i < 32; i++)
				{
					if ((p_sub1.data[index] & mask) != 0)
					{
						index = p_sub1.dataLength;      // to break the outer loop
						break;
					}
					mask <<= 1;
					s++;
				}
			}

			BigInteger t = p_sub1 >> s;

			int bits = thisVal.bitCount();
			BigInteger a = new BigInteger();
			Random rand = new Random();

			for (int round = 0; round < confidence; round++)
			{
				bool done = false;

				while (!done)		// generate a < n
				{
					int testBits = 0;

					// make sure "a" has at least 2 bits
					while (testBits < 2)
						testBits = (int)(rand.NextDouble() * bits);

					a.genRandomBits(testBits, rand);

					int byteLen = a.dataLength;

					// make sure "a" is not 0
					if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
						done = true;
				}

				// check whether a factor exists (fix for version 1.03)
				BigInteger gcdTest = a.gcd(thisVal);
				if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
					return false;

				BigInteger b = a.modPow(t, thisVal);

				bool result = false;

				if (b.dataLength == 1 && b.data[0] == 1)         // a^t mod p = 1
					result = true;

				for (int j = 0; result == false && j < s; j++)
				{
					if (b == p_sub1)         // a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
					{
						result = true;
						break;
					}

					b = (b * b) % thisVal;
				}

				if (result == false)
					return false;
			}
			return true;
		}


		//***********************************************************************
		// Probabilistic prime test based on Solovay-Strassen (Euler Criterion)
		//
		// p is probably prime if for any a < p (a is not multiple of p),
		// a^((p-1)/2) mod p = J(a, p)
		//
		// where J is the Jacobi symbol.
		//
		// Otherwise, p is composite.
		//
		// Returns
		// -------
		// True if "this" is a Euler pseudoprime to randomly chosen
		// bases.  The number of chosen bases is given by the "confidence"
		// parameter.
		//
		// False if "this" is definitely NOT prime.
		//
		//***********************************************************************

		internal bool SolovayStrassenTest(int confidence)
		{
			BigInteger thisVal;
			if ((this.data[maxLength - 1] & 0x80000000) != 0)        // negative
				thisVal = -this;
			else
				thisVal = this;

			if (thisVal.dataLength == 1)
			{
				// test small numbers
				if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
					return false;
				else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
					return true;
			}

			if ((thisVal.data[0] & 0x1) == 0)     // even numbers
				return false;


			int bits = thisVal.bitCount();
			BigInteger a = new BigInteger();
			BigInteger p_sub1 = thisVal - 1;
			BigInteger p_sub1_shift = p_sub1 >> 1;

			Random rand = new Random();

			for (int round = 0; round < confidence; round++)
			{
				bool done = false;

				while (!done)		// generate a < n
				{
					int testBits = 0;

					// make sure "a" has at least 2 bits
					while (testBits < 2)
						testBits = (int)(rand.NextDouble() * bits);

					a.genRandomBits(testBits, rand);

					int byteLen = a.dataLength;

					// make sure "a" is not 0
					if (byteLen > 1 || (byteLen == 1 && a.data[0] != 1))
						done = true;
				}

				// check whether a factor exists (fix for version 1.03)
				BigInteger gcdTest = a.gcd(thisVal);
				if (gcdTest.dataLength == 1 && gcdTest.data[0] != 1)
					return false;

				// calculate a^((p-1)/2) mod p

				BigInteger expResult = a.modPow(p_sub1_shift, thisVal);
				if (expResult == p_sub1)
					expResult = -1;

				// calculate Jacobi symbol
				BigInteger jacob = Jacobi(a, thisVal);

				// if they are different then it is not prime
				if (expResult != jacob)
					return false;
			}

			return true;
		}


		//***********************************************************************
		// Implementation of the Lucas Strong Pseudo Prime test.
		//
		// Let n be an odd number with gcd(n,D) = 1, and n - J(D, n) = 2^s * d
		// with d odd and s >= 0.
		//
		// If Ud mod n = 0 or V2^r*d mod n = 0 for some 0 <= r < s, then n
		// is a strong Lucas pseudoprime with parameters (P, Q).  We select
		// P and Q based on Selfridge.
		//
		// Returns True if number is a strong Lucus pseudo prime.
		// Otherwise, returns False indicating that number is composite.
		//***********************************************************************

		internal bool LucasStrongTest()
		{
			BigInteger thisVal;
			if ((this.data[maxLength - 1] & 0x80000000) != 0)        // negative
				thisVal = -this;
			else
				thisVal = this;

			if (thisVal.dataLength == 1)
			{
				// test small numbers
				if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
					return false;
				else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
					return true;
			}

			if ((thisVal.data[0] & 0x1) == 0)     // even numbers
				return false;

			return LucasStrongTestHelper(thisVal);
		}


		internal bool LucasStrongTestHelper(BigInteger thisVal)
		{
			// Do the test (selects D based on Selfridge)
			// Let D be the first element of the sequence
			// 5, -7, 9, -11, 13, ... for which J(D,n) = -1
			// Let P = 1, Q = (1-D) / 4

			long D = 5, sign = -1, dCount = 0;
			bool done = false;

			while (!done)
			{
				int Jresult = BigInteger.Jacobi(D, thisVal);

				if (Jresult == -1)
					done = true;    // J(D, this) = 1
				else
				{
					if (Jresult == 0 && Math.Abs(D) < thisVal)       // divisor found
						return false;

					if (dCount == 20)
					{
						// check for square
						BigInteger root = thisVal.sqrt();
						if (root * root == thisVal)
							return false;
					}

					//Console.WriteLine(D);
					D = (Math.Abs(D) + 2) * sign;
					sign = -sign;
				}
				dCount++;
			}

			long Q = (1 - D) >> 2;

			BigInteger p_add1 = thisVal + 1;
			int s = 0;

			for (int index = 0; index < p_add1.dataLength; index++)
			{
				uint mask = 0x01;

				for (int i = 0; i < 32; i++)
				{
					if ((p_add1.data[index] & mask) != 0)
					{
						index = p_add1.dataLength;      // to break the outer loop
						break;
					}
					mask <<= 1;
					s++;
				}
			}

			BigInteger t = p_add1 >> s;

			// calculate constant = b^(2k) / m
			// for Barrett Reduction
			BigInteger constant = new BigInteger();

			int nLen = thisVal.dataLength << 1;
			constant.data[nLen] = 0x00000001;
			constant.dataLength = nLen + 1;

			constant = constant / thisVal;

			BigInteger[] lucas = LucasSequenceHelper(1, Q, t, thisVal, constant, 0);
			bool isPrime = false;

			if ((lucas[0].dataLength == 1 && lucas[0].data[0] == 0) ||
			   (lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
			{
				// u(t) = 0 or V(t) = 0
				isPrime = true;
			}

			for (int i = 1; i < s; i++)
			{
				if (!isPrime)
				{
					// doubling of index
					lucas[1] = thisVal.BarrettReduction(lucas[1] * lucas[1], thisVal, constant);
					lucas[1] = (lucas[1] - (lucas[2] << 1)) % thisVal;

					if ((lucas[1].dataLength == 1 && lucas[1].data[0] == 0))
						isPrime = true;
				}

				lucas[2] = thisVal.BarrettReduction(lucas[2] * lucas[2], thisVal, constant);     //Q^k
			}


			if (isPrime)     // additional checks for composite numbers
			{
				// If n is prime and gcd(n, Q) == 1, then
				// Q^((n+1)/2) = Q * Q^((n-1)/2) is congruent to (Q * J(Q, n)) mod n

				BigInteger g = thisVal.gcd(Q);
				if (g.dataLength == 1 && g.data[0] == 1)         // gcd(this, Q) == 1
				{
					if ((lucas[2].data[maxLength - 1] & 0x80000000) != 0)
						lucas[2] += thisVal;

					BigInteger temp = (Q * BigInteger.Jacobi(Q, thisVal)) % thisVal;
					if ((temp.data[maxLength - 1] & 0x80000000) != 0)
						temp += thisVal;

					if (lucas[2] != temp)
						isPrime = false;
				}
			}

			return isPrime;
		}


		//***********************************************************************
		// Determines whether a number is probably prime, using the Rabin-Miller's
		// test.  Before applying the test, the number is tested for divisibility
		// by primes < 2000
		//
		// Returns true if number is probably prime.
		//***********************************************************************

		internal bool isProbablePrime(int confidence)
		{
			BigInteger thisVal;
			if ((this.data[maxLength - 1] & 0x80000000) != 0)        // negative
				thisVal = -this;
			else
				thisVal = this;


			// test for divisibility by primes < 2000
			for (int p = 0; p < primesBelow2000.Length; p++)
			{
				BigInteger divisor = primesBelow2000[p];

				if (divisor >= thisVal)
					break;

				BigInteger resultNum = thisVal % divisor;
				if (resultNum.IntValue() == 0)
				{

					return false;
				}
			}

			if (thisVal.RabinMillerTest(confidence))
				return true;
			else
			{
				return false;
			}
		}


		//***********************************************************************
		// Determines whether this BigInteger is probably prime using a
		// combination of base 2 strong pseudoprime test and Lucas strong
		// pseudoprime test.
		//
		// The sequence of the primality test is as follows,
		//
		// 1) Trial divisions are carried out using prime numbers below 2000.
		//    if any of the primes divides this BigInteger, then it is not prime.
		//
		// 2) Perform base 2 strong pseudoprime test.  If this BigInteger is a
		//    base 2 strong pseudoprime, proceed on to the next step.
		//
		// 3) Perform strong Lucas pseudoprime test.
		//
		// Returns True if this BigInteger is both a base 2 strong pseudoprime
		// and a strong Lucas pseudoprime.
		//
		// For a detailed discussion of this primality test, see [6].
		//
		//***********************************************************************

		internal bool isProbablePrime()
		{
			BigInteger thisVal;
			if ((this.data[maxLength - 1] & 0x80000000) != 0)        // negative
				thisVal = -this;
			else
				thisVal = this;

			if (thisVal.dataLength == 1)
			{
				// test small numbers
				if (thisVal.data[0] == 0 || thisVal.data[0] == 1)
					return false;
				else if (thisVal.data[0] == 2 || thisVal.data[0] == 3)
					return true;
			}

			if ((thisVal.data[0] & 0x1) == 0)     // even numbers
				return false;


			// test for divisibility by primes < 2000
			for (int p = 0; p < primesBelow2000.Length; p++)
			{
				BigInteger divisor = primesBelow2000[p];

				if (divisor >= thisVal)
					break;

				BigInteger resultNum = thisVal % divisor;
				if (resultNum.IntValue() == 0)
				{
					return false;
				}
			}

			// Perform BASE 2 Rabin-Miller Test
			// calculate values of s and t
			BigInteger p_sub1 = thisVal - (new BigInteger(1));
			int s = 0;

			for (int index = 0; index < p_sub1.dataLength; index++)
			{
				uint mask = 0x01;

				for (int i = 0; i < 32; i++)
				{
					if ((p_sub1.data[index] & mask) != 0)
					{
						index = p_sub1.dataLength;      // to break the outer loop
						break;
					}
					mask <<= 1;
					s++;
				}
			}

			BigInteger t = p_sub1 >> s;

			int bits = thisVal.bitCount();
			BigInteger a = 2;

			// b = a^t mod p
			BigInteger b = a.modPow(t, thisVal);

			bool result = false;

			if (b.dataLength == 1 && b.data[0] == 1)         // a^t mod p = 1
				result = true;

			for (int j = 0; result == false && j < s; j++)
			{
				if (b == p_sub1)         // a^((2^j)*t) mod p = p-1 for some 0 <= j <= s-1
				{
					result = true;
					break;
				}

				b = (b * b) % thisVal;
			}

			//if number is strong pseudoprime to base 2, then do a strong lucas test
			if (result)
			result = LucasStrongTestHelper(thisVal);

			return result;
		}


		//***********************************************************************
		// Returns the lowest 4 bytes of the BigInteger as an int.
		//***********************************************************************

		internal int IntValue()
		{
			return (int)data[0];
		}


		//***********************************************************************
		// Returns the lowest 8 bytes of the BigInteger as a long.
		//***********************************************************************

		internal long LongValue()
		{
			long val = 0;

			val = (long)data[0];
			try
			{       // exception if maxLength = 1
				val |= (long)data[1] << 32;
			}
			catch (Exception)
			{
				if ((data[0] & 0x80000000) != 0) // negative
					val = (int)data[0];
			}

			return val;
		}


		//***********************************************************************
		// Computes the Jacobi Symbol for a and b.
		// Algorithm adapted from [3] and [4] with some optimizations
		//***********************************************************************

		internal static int Jacobi(BigInteger a, BigInteger b)
		{
			// Jacobi defined only for odd integers
			if ((b.data[0] & 0x1) == 0)
				throw (new ArgumentException("Jacobi defined only for odd integers."));

			if (a >= b) a %= b;
			if (a.dataLength == 1 && a.data[0] == 0) return 0;  // a == 0
			if (a.dataLength == 1 && a.data[0] == 1) return 1;  // a == 1

			if (a < 0)
			{
				if ((((b - 1).data[0]) & 0x2) == 0)       //if( (((b-1) >> 1).data[0] & 0x1) == 0)
					return Jacobi(-a, b);
				else
					return -Jacobi(-a, b);
			}

			int e = 0;
			for (int index = 0; index < a.dataLength; index++)
			{
				uint mask = 0x01;

				for (int i = 0; i < 32; i++)
				{
					if ((a.data[index] & mask) != 0)
					{
						index = a.dataLength;      // to break the outer loop
						break;
					}
					mask <<= 1;
					e++;
				}
			}

			BigInteger a1 = a >> e;

			int s = 1;
			if ((e & 0x1) != 0 && ((b.data[0] & 0x7) == 3 || (b.data[0] & 0x7) == 5))
				s = -1;

			if ((b.data[0] & 0x3) == 3 && (a1.data[0] & 0x3) == 3)
				s = -s;

			if (a1.dataLength == 1 && a1.data[0] == 1)
				return s;
			else
				return (s * Jacobi(b % a1, a1));
		}


		//***********************************************************************
		// Generates a positive BigInteger that is probably prime.
		//***********************************************************************

		internal static BigInteger genPseudoPrime(int bits, int confidence, Random rand)
		{
			BigInteger result = new BigInteger();
			bool done = false;

			while (!done)
			{
				result.genRandomBits(bits, rand);
				result.data[0] |= 0x01;		// make it odd

				// prime test
				done = result.isProbablePrime(confidence);
			}
			return result;
		}

		//***********************************************************************
		// Generates a positive BigInteger that is probably prime.
		// Overloaded to use the isProbablePrime method with no confidence value
		//***********************************************************************

		internal static BigInteger genPseudoPrime(int bits, Random rand)
		{
			BigInteger result = new BigInteger();
			bool done = false;

			while (!done)
			{
				result.genRandomBits(bits, rand);
				result.data[0] |= 0x01;		// make it odd

				// prime test
				done = result.isProbablePrime();
			}
			return result;
		}


		//***********************************************************************
		// Generates a random number with the specified number of bits such
		// that gcd(number, this) = 1
		//***********************************************************************

		internal BigInteger genCoPrime(int bits, Random rand)
		{
			bool done = false;
			BigInteger result = new BigInteger();

			while (!done)
			{
				result.genRandomBits(bits, rand);

				// gcd test
				BigInteger g = result.gcd(this);
				if (g.dataLength == 1 && g.data[0] == 1)
					done = true;
			}

			return result;
		}


		//***********************************************************************
		// Returns the modulo inverse of this.  Throws ArithmeticException if
		// the inverse does not exist.  (i.e. gcd(this, modulus) != 1)
		//***********************************************************************

		internal BigInteger modInverse(BigInteger modulus)
		{
			BigInteger[] p = { 0, 1 };
			BigInteger[] q = new BigInteger[2];    // quotients
			BigInteger[] r = { 0, 0 };             // remainders

			int step = 0;

			BigInteger a = modulus;
			BigInteger b = this;

			while (b.dataLength > 1 || (b.dataLength == 1 && b.data[0] != 0))
			{
				BigInteger quotient = new BigInteger();
				BigInteger remainder = new BigInteger();

				if (step > 1)
				{
					BigInteger pval = (p[0] - (p[1] * q[0])) % modulus;
					p[0] = p[1];
					p[1] = pval;
				}

				if (b.dataLength == 1)
					singleByteDivide(a, b, quotient, remainder);
				else
					multiByteDivide(a, b, quotient, remainder);

				q[0] = q[1];
				r[0] = r[1];
				q[1] = quotient; r[1] = remainder;

				a = b;
				b = remainder;

				step++;
			}

			if (r[0].dataLength > 1 || (r[0].dataLength == 1 && r[0].data[0] != 1))
				throw (new ArithmeticException("No inverse!"));

			BigInteger result = ((p[0] - (p[1] * q[0])) % modulus);

			if ((result.data[maxLength - 1] & 0x80000000) != 0)
				result += modulus;  // get the least positive modulus

			return result;
		}


		//***********************************************************************
		// Returns the position of the most significant bit in the BigInteger.
		//
		// Eg.  The result is 0, if the value of BigInteger is 0...0000 0000
		//      The result is 1, if the value of BigInteger is 0...0000 0001
		//      The result is 2, if the value of BigInteger is 0...0000 0010
		//      The result is 2, if the value of BigInteger is 0...0000 0011
		//
		//***********************************************************************

		internal int bitCount()
		{
			while (dataLength > 1 && data[dataLength - 1] == 0)
				dataLength--;

			uint value = data[dataLength - 1];
			uint mask = 0x80000000;
			int bits = 32;

			while (bits > 0 && (value & mask) == 0)
			{
				bits--;
				mask >>= 1;
			}
			bits += ((dataLength - 1) << 5);

			return bits;
		}

		internal int bitCountRaw()
		{
			uint value = data[dataLength - 1];
			int bits = 32;

			bits += ((dataLength - 1) << 5);

			return bits;
		}


		//***********************************************************************
		// Returns the value of the BigInteger as a byte array.  The lowest
		// index contains the MSB.
		//***********************************************************************

		internal byte[] getBytes()
		{
			int numBits = bitCount();
			byte[] result = null;
			if (numBits == 0)
			{
				result = new byte[1];
				result[0] = 0;
			}
			else
			{
				int numBytes = numBits >> 3;
				if ((numBits & 0x7) != 0) numBytes++;
				result = new byte[numBytes];


				int numBytesInWord = numBytes & 0x3;
				if (numBytesInWord == 0) numBytesInWord = 4;
				int pos = 0;
				for (int i = dataLength - 1; i >= 0; i--)
				{
					uint val = data[i];
					for (int j = numBytesInWord - 1; j >= 0; j--)
					{
						result[pos + j] = (byte)(val & 0xFF);
						val >>= 8;
					}
					pos += numBytesInWord;
					numBytesInWord = 4;
				}
			} return result;
		}

		public byte[] getBytesRaw()
		{
			int numBits = bitCountRaw();
			byte[] result = null;
			if (numBits == 0)
			{
				result = new byte[1];
				result[0] = 0;
			}
			else
			{
				int numBytes = numBits >> 3;
				if ((numBits & 0x7) != 0) numBytes++;
				result = new byte[numBytes];


				int numBytesInWord = numBytes & 0x3;
				if (numBytesInWord == 0) numBytesInWord = 4;
				int pos = 0;
				for (int i = dataLength - 1; i >= 0; i--)
				{
					uint val = data[i];
					for (int j = numBytesInWord - 1; j >= 0; j--)
					{
						result[pos + j] = (byte)(val & 0xFF);
						val >>= 8;
					}
					pos += numBytesInWord;
					numBytesInWord = 4;
				}
			} return result;
		}


		//***********************************************************************
		// Sets the value of the specified bit to 1
		// The Least Significant Bit position is 0.
		//***********************************************************************

		internal void setBit(uint bitNum)
		{
			uint bytePos = bitNum >> 5;             // divide by 32
			byte bitPos = (byte)(bitNum & 0x1F);    // get the lowest 5 bits

			uint mask = (uint)1 << bitPos;
			this.data[bytePos] |= mask;

			if (bytePos >= this.dataLength)
				this.dataLength = (int)bytePos + 1;
		}


		//***********************************************************************
		// Sets the value of the specified bit to 0
		// The Least Significant Bit position is 0.
		//***********************************************************************

		internal void unsetBit(uint bitNum)
		{
			uint bytePos = bitNum >> 5;

			if (bytePos < this.dataLength)
			{
				byte bitPos = (byte)(bitNum & 0x1F);

				uint mask = (uint)1 << bitPos;
				uint mask2 = 0xFFFFFFFF ^ mask;

				this.data[bytePos] &= mask2;

				if (this.dataLength > 1 && this.data[this.dataLength - 1] == 0)
					this.dataLength--;
			}
		}


		//***********************************************************************
		// Returns a value that is equivalent to the integer square root
		// of the BigInteger.
		//
		// The integer square root of "this" is defined as the largest integer n
		// such that (n * n) <= this
		//
		//***********************************************************************

		internal BigInteger sqrt()
		{
			uint numBits = (uint)this.bitCount();

			if ((numBits & 0x1) != 0)        // odd number of bits
				numBits = (numBits >> 1) + 1;
			else
				numBits = (numBits >> 1);

			uint bytePos = numBits >> 5;
			byte bitPos = (byte)(numBits & 0x1F);

			uint mask;

			BigInteger result = new BigInteger();
			if (bitPos == 0)
				mask = 0x80000000;
			else
			{
				mask = (uint)1 << bitPos;
				bytePos++;
			}
			result.dataLength = (int)bytePos;

			for (int i = (int)bytePos - 1; i >= 0; i--)
			{
				while (mask != 0)
				{
					// guess
					result.data[i] ^= mask;

					// undo the guess if its square is larger than this
					if ((result * result) > this)
						result.data[i] ^= mask;

					mask >>= 1;
				}
				mask = 0x80000000;
			}
			return result;
		}


		//***********************************************************************
		// Returns the k_th number in the Lucas Sequence reduced modulo n.
		//
		// Uses index doubling to speed up the process.  For example, to calculate V(k),
		// we maintain two numbers in the sequence V(n) and V(n+1).
		//
		// To obtain V(2n), we use the identity
		//      V(2n) = (V(n) * V(n)) - (2 * Q^n)
		// To obtain V(2n+1), we first write it as
		//      V(2n+1) = V((n+1) + n)
		// and use the identity
		//      V(m+n) = V(m) * V(n) - Q * V(m-n)
		// Hence,
		//      V((n+1) + n) = V(n+1) * V(n) - Q^n * V((n+1) - n)
		//                   = V(n+1) * V(n) - Q^n * V(1)
		//                   = V(n+1) * V(n) - Q^n * P
		//
		// We use k in its binary expansion and perform index doubling for each
		// bit position.  For each bit position that is set, we perform an
		// index doubling followed by an index addition.  This means that for V(n),
		// we need to update it to V(2n+1).  For V(n+1), we need to update it to
		// V((2n+1)+1) = V(2*(n+1))
		//
		// This function returns
		// [0] = U(k)
		// [1] = V(k)
		// [2] = Q^n
		//
		// Where U(0) = 0 % n, U(1) = 1 % n
		//       V(0) = 2 % n, V(1) = P % n
		//***********************************************************************

		internal static BigInteger[] LucasSequence(BigInteger P, BigInteger Q,
												 BigInteger k, BigInteger n)
		{
			if (k.dataLength == 1 && k.data[0] == 0)
			{
				BigInteger[] result = new BigInteger[3];

				result[0] = 0; result[1] = 2 % n; result[2] = 1 % n;
				return result;
			}

			// calculate constant = b^(2k) / m
			// for Barrett Reduction
			BigInteger constant = new BigInteger();

			int nLen = n.dataLength << 1;
			constant.data[nLen] = 0x00000001;
			constant.dataLength = nLen + 1;

			constant = constant / n;

			// calculate values of s and t
			int s = 0;

			for (int index = 0; index < k.dataLength; index++)
			{
				uint mask = 0x01;

				for (int i = 0; i < 32; i++)
				{
					if ((k.data[index] & mask) != 0)
					{
						index = k.dataLength;      // to break the outer loop
						break;
					}
					mask <<= 1;
					s++;
				}
			}

			BigInteger t = k >> s;

			//Console.WriteLine("s = " + s + " t = " + t);
			return LucasSequenceHelper(P, Q, t, n, constant, s);
		}


		//***********************************************************************
		// Performs the calculation of the kth term in the Lucas Sequence.
		// For details of the algorithm, see reference [9].
		//
		// k must be odd.  i.e LSB == 1
		//***********************************************************************

		private static BigInteger[] LucasSequenceHelper(BigInteger P, BigInteger Q,
														BigInteger k, BigInteger n,
														BigInteger constant, int s)
		{
			BigInteger[] result = new BigInteger[3];

			if ((k.data[0] & 0x00000001) == 0)
				throw (new ArgumentException("Argument k must be odd."));

			int numbits = k.bitCount();
			uint mask = (uint)0x1 << ((numbits & 0x1F) - 1);

			// v = v0, v1 = v1, u1 = u1, Q_k = Q^0

			BigInteger v = 2 % n, Q_k = 1 % n,
					   v1 = P % n, u1 = Q_k;
			bool flag = true;

			for (int i = k.dataLength - 1; i >= 0; i--)     // iterate on the binary expansion of k
			{
				//Console.WriteLine("round");
				while (mask != 0)
				{
					if (i == 0 && mask == 0x00000001)        // last bit
						break;

					if ((k.data[i] & mask) != 0)             // bit is set
					{
						// index doubling with addition

						u1 = (u1 * v1) % n;

						v = ((v * v1) - (P * Q_k)) % n;
						v1 = n.BarrettReduction(v1 * v1, n, constant);
						v1 = (v1 - ((Q_k * Q) << 1)) % n;

						if (flag)
							flag = false;
						else
							Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);

						Q_k = (Q_k * Q) % n;
					}
					else
					{
						// index doubling
						u1 = ((u1 * v) - Q_k) % n;

						v1 = ((v * v1) - (P * Q_k)) % n;
						v = n.BarrettReduction(v * v, n, constant);
						v = (v - (Q_k << 1)) % n;

						if (flag)
						{
							Q_k = Q % n;
							flag = false;
						}
						else
							Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);
					}

					mask >>= 1;
				}
				mask = 0x80000000;
			}

			// at this point u1 = u(n+1) and v = v(n)
			// since the last bit always 1, we need to transform u1 to u(2n+1) and v to v(2n+1)

			u1 = ((u1 * v) - Q_k) % n;
			v = ((v * v1) - (P * Q_k)) % n;
			if (flag)
				flag = false;
			else
				Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);

			Q_k = (Q_k * Q) % n;


			for (int i = 0; i < s; i++)
			{
				// index doubling
				u1 = (u1 * v) % n;
				v = ((v * v) - (Q_k << 1)) % n;

				if (flag)
				{
					Q_k = Q % n;
					flag = false;
				}
				else
					Q_k = n.BarrettReduction(Q_k * Q_k, n, constant);
			}

			result[0] = u1;
			result[1] = v;
			result[2] = Q_k;

			return result;
		}

	}
}